A Spectral–Entropy Framework for the Mersenne Infinitude Problem: Multi-scale Density–0 Rigidity and Transduction Diagnostics

## Overview This preprint presents a **windowed operator–variational program** for the *Mersenne infinitude problem* (Mersenne primes \(M_p=2^p-1\)). The paper is written to **isolate, with surgical clarity, where genuinely arithmetic input is required**, and to provide a rigorous “density–0 closure” module once that input is supplied. The core philosophy is modular: - a kernel/Markov-operator layer provides **mixing diagnostics** (spectral gap / LSI) on exponent windows,- an entropy–rigidity functional provides a **logarithmic density–0 closure mechanism** for Mersenne-empty windows,- the only Mersenne-specific obstacle is separated as a **transduction hypothesis**, whose threshold can be supported empirically. --- ## What is new in v0.8 - **Barrier section added (methodological necessity).** - A **logical non-certifiability barrier** is recorded: band-limited (finite factor-test) diagnostics cannot certify primality. - An **internal necessity theorem** clarifies that a transduction-type input (or a stronger surrogate) is required for unconditional closure within this pipeline.- **Multi-scale Mask A is elevated as the main exceptional-window mechanism,** tying the energy-gap target to deficits across multiple disjoint factor bands (stronger than a single cutoff).- **Exploratory empirical validation is strengthened** (Appendix C): - pilot ROC/AUC and calibration/reliability diagrams on a bounded range (e.g. \(p\le 10^7\)), - a GIMPS catalogue contrast on known Mersenne-prime exponents as a screening stress test, - robustness/negative-control checks for the proxy diagnostic. These additions are intended to **increase transparency and falsifiability** while avoiding over-claiming. --- ## Main technical components (high level) ### Windowed exponent modelWe work on prime exponents \(p\) in variable windows\[I_T=[T,\;T+T^\alpha],\qquad \alpha\in(0,1),\]with a finite probability space \((\mathcal{O}_T,m_T)\) of windowed exponents. ### Incidence kernel and lazy MarkovizationWe introduce factor-admissible prime bands and construct a symmetric band-limited incidence kernel\(K^{(1)}_{M,T}\). A lazy Markovization \(\widetilde P_{M,T}\) is used to define a mixing diagnostic(spectral gap / LSI). ### Sieve-to-mixing bridge (conditional)The paper formulates a conditional bridge: BV/large-sieve-type control on incidence degrees/overlapsimplies conductance bounds and hence polylogarithmic mixing diagnostics (gap/LSI) for \(\widetilde P_{M,T}\). ### Entropy–rigidity functional and density–0 closure (rigorous module)For a measurable mask \(E_T\subset \mathcal{O}_T\), we study an entropy–rigidity functional of the form\[\widetilde{\mathcal{G}}_{M,T}[\rho]=\mathcal{H}(\rho|m_T)+\lambda_1\|\widetilde P_{M,T}\rho-\rho\|_2^2+\lambda_2\int_{\mathcal{O}_T}\rho^2\mathbf{1}_{E_T}\,dm_T.\]Assuming (i) a mixing diagnostic and (ii) an energy-gap premise tied to an explicit exceptional-window mask,we prove that the set of Mersenne-empty windows has **logarithmic density \(0\)**. ### Transduction (open target)The step “diagnostic \(\Rightarrow\) existence of a Mersenne prime in the window” is isolated as a**transduction hypothesis**. v0.8 emphasizes that this is the only genuinely Mersenne-specific obstacleremaining for unconditional closure in the present framework. --- ## Numerical appendix (exploratory, not a proof substitute) Numerics are included to validate diagnostic behavior (gap estimators, band stability, robustness ofrescaled quantities, and empirical threshold calibration). They are **not** used as a replacement forthe transduction or energy-gap steps. --- ## Scope and non-claims - v0.8 does **not** claim an unconditional proof of infinitely many Mersenne primes from classical sieve bounds alone.- The barrier section is **methodological**, not metamathematical: it does **not** claim independence from ZFC/PA.- Mellin–trace identities, when used, are treated only as bookkeeping for windowed diagonal observables,not as a standalone analytic-continuation route to infinitude. --- ## Suggested citation Lee, Byoungwoo. *A Windowed Operator–Variational Framework for the Mersenne Infinitude Problem:Multi-scale Density–0 Rigidity, Transduction Diagnostics, and Methodological Barriers* (v0.8). Zenodo, 2026. --- ## Keywords Mersenne primes; analytic number theory; sieve methods; large sieve; Bombieri–Vinogradov;Markov chains; conductance; spectral gap; logarithmic Sobolev inequality; entropy dissipation;rigidity functional; logarithmic density; variable windows; ROC/AUC; calibration; reproducibility.